The design of radio-frequency power amplifiers for communications applications often involves a trade-off between linearity and efficiency. Power amplifiers are typically most efficient when operated at or near their saturation point. However, the response of the amplifier at or near the point of saturation is non-linear. Generally speaking, when operating in the high-efficiency range, a power amplifier's response exhibits a nonlinear response and memory effects.
One way to improve a power amplifier's efficiency and its overall linearity is to digitally pre-distort the input to the power amplifier to compensate for the distortion introduced by the power amplifier. In effect, the input signal is adjusted in anticipation of the distortion to be introduced by the power amplifier, so that the output signal is largely free of distortion products. Generally, digital pre-distortion is applied to the signal at baseband frequencies, i.e., before the signal is up-converted to radio frequencies.
Thus, for a power amplifier to achieve high efficiency, the power amplifier is operated in a non-linear region. This causes distortion of the input signal and broadening of the bandwidth of the input signal. To compensate for the distortion of the signal introduced by the power amplifier, the input signal is first passed through a pre-distorter that pre-distorts the input signal. A typical pre-distorter is itself non-linear, having a non-linearity that compensates for the non-linearity of the pre-distorter. To illustrate, a power amplifier may exhibit first and third order effects characterized by a polynomial function of the input that may be written for third order non-linearities as:y=fNL-IM3(x)=a1x+a3x3  (01)where x is the input signal and a3 is much less than a1. The function f is the response of the power amplifier to the input x and the subscript NL-IM3 denotes non-linearity up to order three. To compensate for the distortion introduced by the power amplifier, a pre-distorter may have a response that is a polynomial function of the input:z=fPD-IM3(x)=b1x+b3x3  (02)Substituting equation (02) into equation (01) leads to:y=fNL-IM3(fPD-IM39x))=a1b1x+(a1b3+a3b13)x3+O(x5)  (03)where O(x5) are terms of 5th order or higher. By appropriate selection of the coefficients b1 and b3, the third order term may be removed at the expense of creating higher order terms of less significant magnitude. The solution to achieve this is given by:b3=−a3b13/a1  (04)Without loss of generality, assume that a1=b1=1. Then the solution to compensate for third order distortions is:b3=−a3  (05)This simple illustration is for third order non-linearities. For higher order non-linearities, the same approach may be taken to cancel the higher order terms. Thus, the pre-distorter is a non-linear device that compensates for the distortion caused by the power amplifier.
These techniques can be quite beneficial in improving the overall performance of a transmitter system, in terms of both linearity and efficiency. Furthermore, these techniques can be relatively inexpensive, due to the digital implementation of the pre-distorter. In fact, with the availability of these techniques, power amplifiers may be designed in view of more relaxed linearity requirements than would otherwise be permissible, thus potentially reducing the costs of the overall system.
The bandwidth of the pre-distorter must be wider than the bandwidth of the input signal depending on the order of inter-modulation to be compensated by the pre-distorter. For example, for third order inter-modulations, the pre-distorted signal occupies about three times the bandwidth of the input signal to the pre-distorter. For fifth order inter-modulations, the pre-distorted signal occupies about 5 times the bandwidth of the input signal. Higher bandwidth implies that the sampling rate of the pre-distorted signal must be higher than the sampling rate of the sampled baseband signal to avoid aliasing.
FIG. 1 shows a known dual band power amplifier system 10 with pre-distorters 12 and 14 and adaptors 16 and 18. The adaptors 16 and 18 determine pre-distorter coefficients for pre-distorters 12 and 14, respectively. Each pre-distorter 12 and 14 receives a signal from a respective source 20 and 22 that is up-sampled by a corresponding up-sampler 24 or 26. These inputs to the pre-distorters 12 and 14 are reference signals that are also input to the adaptors 16 and 18, respectively. The outputs of the pre-distorters 12 and 14 are up-converted in frequency to a first carrier ejωcarrier1 and a second carrier ejωcarrier2, respectively, via a corresponding up-conversion unit 28 or 30. The outputs of the up-conversion units 28 and 30 are summed by an adder 32. The output of the adder 32 may be quadrature-modulated by a quadrature modulation unit 34. The output of the quadrature modulation unit 34 is input to the power amplifier 36. The output of the power amplifier 36 is input to one of two filters 38 and 40 of an observation receiver 11. The outputs of these filters are down-converted and demodulated by down conversion/demodulation units 42 and 44, respectively. The outputs of units 42 and 44 are observation signals that are input to the adaptors 16 and 18, respectively.
FIG. 2 shows a known alternative power amplifier system 60 using only a single adaptor 58. The observation signal inputs to the adaptor 58 are tuned to IF1 and IF2 by corresponding tuners 46 and 48 of the observation receiver 61. Similarly, the reference signal inputs to the adaptor 58 are tuned to IF1 and IF2 by tuners 50 and 52. In the configuration of FIG. 2, the adaptation parameters produced by the adaptor 58 are used by both pre-distorters 12 and 14. The basis functions in each pre-distorter, for fifth order non-linearity compensation are of the form:y1=x1+x1|x1|2+x1|x2|2+x1|x1|4+x1|x1|2|x2|2+x1|x2|4  (06)y2=x2+x2|x2|2+x2|x1|2+x2|x2|4+x2|x2|2|x1|2+x2|x1|4  (07)where y1 and y2 represent the output of the pre-distorters 12 and 14, respectively, and x1 and x2 are the baseband input signals input to the pre-distorters 12 and 14, respectively. The error that the adaptor seeks to minimize is given by:
                    e        =                              (                                                            x                  1                                ⁢                                  ⅇ                                      j                    ⁢                                                                                  ⁢                                          ω                                              IF                        ⁢                                                                                                  ⁢                        1                                                                                                        +                                                x                  2                                ⁢                                  ⅇ                                      j                    ⁢                                                                                  ⁢                                          ω                                              IF                        ⁢                                                                                                  ⁢                        2                                                                                                                  )                    -                      (                                                                                                                                                        (                                                                                                                    c                                0                                                            ⁢                                                              x                                1                                                                                      +                                                                                          c                                1                                                            ⁢                                                              x                                1                                                            ⁢                                                                                                                                                                      x                                    1                                                                                                                                    2                                                                                      +                                                                                          c                                2                                                            ⁢                                                              x                                1                                                            ⁢                                                                                                                                                                      x                                    2                                                                                                                                    2                                                                                      +                                                                                          c                                3                                                            ⁢                                                              x                                1                                                            ⁢                                                                                                                                                                      x                                    1                                                                                                                                    4                                                                                      +                                                                                                                                                                                                                                                                                                                                                        c                                    4                                                                    ⁢                                                                      x                                    1                                                                    ⁢                                                                                                                                                                                          x                                        1                                                                                                                                                    2                                                                    ⁢                                                                                                                                                                                          x                                        2                                                                                                                                                    2                                                                                                  +                                                                                                      c                                    5                                                                    ⁢                                                                      x                                    1                                                                    ⁢                                                                                                                                                                                          x                                        2                                                                                                                                                    4                                                                                                                              )                                                        ⁢                                                          ⅇ                                                                                                j                                  ⁢                                                                                                                                          ⁢                                                                      ω                                                                          IF                                      ⁢                                                                                                                                                          ⁢                                      1                                                                                                                                      ⁢                                                                                                                                                                                                                +                                                                                                                                                                                                                                                              (                                                                                                                    c                                0                                                            ⁢                                                              x                                2                                                                                      +                                                                                          c                                1                                                            ⁢                                                              x                                2                                                            ⁢                                                                                                                                                                      x                                    2                                                                                                                                    2                                                                                      +                                                                                          c                                2                                                            ⁢                                                              x                                2                                                            ⁢                                                                                                                                                                      x                                    1                                                                                                                                    2                                                                                      +                                                                                          c                                3                                                            ⁢                                                              x                                2                                                            ⁢                                                                                                                                                                      x                                    2                                                                                                                                    4                                                                                      +                                                                                                                                                                                                                                                                                                                    c                                  4                                                                ⁢                                                                  x                                  2                                                                ⁢                                                                                                                                                                                x                                      2                                                                                                                                            2                                                                ⁢                                                                                                                                                                                x                                      1                                                                                                                                            2                                                                                            +                                                                                                c                                  5                                                                ⁢                                                                  x                                  2                                                                ⁢                                                                                                                                                                                x                                      1                                                                                                                                            4                                                                                                                      )                                                    ⁢                                                      ⅇ                                                          j                              ⁢                                                                                                                          ⁢                                                              ω                                                                  IF                                  ⁢                                                                                                                                          ⁢                                  2                                                                                                                                                                                                                                                              )                                              (        08        )            where c0-cn−1 are the coefficients generated by the adaptor 58.
FIG. 3 shows fifth order pre-distortion of signals received from sources 20 and 22 that each have an original bandwidth B. To avoid aliasing of these two signals when added in adder 56, the signals must be sampled at the Nyquist rate of at least 5×2×B and tuned to an intermediate frequency IF equal to +/−B/2. FIG. 4 shows fifth order distortion of signals received from the sources 20 and 22 that have an original bandwidth of B1 and B2, respectively. To avoid aliasing, the signals must be un-converted to respective intermediate frequencies IF1 and IF2, given by:
                              IF          ⁢                                          ⁢          1                =                                            -                                                5                  ⁢                                      (                                                                  B                        1                                            +                                              B                        2                                                              )                                                  2                                      +                                          5                2                            ⁢                              B                1                                              =                                    -                              5                2                                      ⁢                          B              2                                                          (        09        )                                          IF          ⁢                                          ⁢          2                =                                                            5                ⁢                                  (                                                            B                      1                                        +                                          B                      2                                                        )                                            2                        -                                          5                2                            ⁢                              B                2                                              =                                    5              2                        ⁢                          B              1                                                          (        10        )            and the sampling rate must be chosen as:fs>5×(b1+B2)  (11)
The embodiment of FIG. 2 requires a high sampling rate inside the adaptor 58. A very high sampling rate is undesirable since a high clock rate may not be available within the system, and/or is more costly to implement, consumes additional power, etc.